Stable-Matching Voronoi Diagrams: Combinatorial Complexity and Algorithms

TL;DR

This paper analyzes the combinatorial complexity and algorithms for stable-matching Voronoi diagrams, which assign points to sites with capacity constraints, providing bounds and a construction algorithm.

Contribution

It establishes near-quadratic bounds on the complexity of stable-matching Voronoi diagrams and introduces an algorithm for their construction.

Findings

Stable-matching Voronoi diagrams have O(n^{2+ε}) faces and edges.

There exist diagrams with Θ(n^2) faces and edges.

An algorithm with O(n^3 log n + n^2 f(n)) runtime for construction.

Abstract

We study algorithms and combinatorial complexity bounds for \emph{stable-matching Voronoi diagrams}, where a set, $S$, of $n$ point sites in the plane determines a stable matching between the points in $\mathbb{R}^2$ and the sites in $S$ such that (i) the points prefer sites closer to them and sites prefer points closer to them, and (ii) each site has a quota or "appetite" indicating the area of the set of points that can be matched to it. Thus, a stable-matching Voronoi diagram is a solution to the well-known post office problem with the added (realistic) constraint that each post office has a limit on the size of its jurisdiction. Previous work on the stable-matching Voronoi diagram provided existence and uniqueness proofs, but did not analyze its combinatorial or algorithmic complexity. In this paper, we show that a stable-matching Voronoi diagram of $n$ point sites has $O(n^{2+\varepsilon})$ faces and edges, for any $\varepsilon>0$, and show that this bound is almost tight by giving a family of diagrams with $\Theta(n^2)$ faces and edges. We also provide a discrete algorithm for constructing it in $O(n^3\log n+n^2f(n))$ time in the real-RAM model of computation, where $f(n)$ is the runtime of a geometric primitive (which we define) that can be approximated numerically, but cannot, in general, be performed exactly in an algebraic model of computation. We show, however, how to compute the geometric primitive exactly for polygonal convex distance functions.